What will happen to the inductance of a solenoid
a. when the number of turns and the length are double keeping the area of cross section same?
b. when the air inside the solenoid is replaced by iron of relative permeability `mu_(r)`?

Explanation

In case of a solenoid as `B = mu_(0)nI, f = B(nlA) = mu_(0)n^(2)lAI` and
hence `L = (phi)/(I) = mu_(0)n^(2)lA = mu_(0) (N^(2))/(l)A`.
a. When `N` and `l` are double,
`L' = mu_(0) ((2N)^(2))/(2l) A = 2 mu_(0)(N^(2))/(l) A = 2L`
i.e., inductance of the solenoid will be doubled.
b. When air is replaced by iron, `mu_(0)` will change to `mu`, so that
`L' = mu n^(2)lA` and hence, `(L')/(L) = (mu)/(mu_(0)) = mu_(r)`
i.e., `L' = mu_(r) L`
So, the inductance will become`mu_(r)` times of its intial value.